1. Abstract

• EXERCISES
  4.19. Study the behavior of our model for Hyperion for different initial conditions. Estimate the Lyapunov exponent from calculations of Δθ, such as those shown in Figure 4.19. Examine how this exponent varies as a function of the eccentricity of the orbit. <br />
  4.20. Our results for the divergence of the two trajectories θ1(t) and θ2(t) in the chaotic regime, shown on the right in Figure 4.19, are complicated by the way we dealt with the angle θ. In Figure 4.19 we followed the practice employed in Chapter 3 and restricted θ to the range -π to +π, since angles ouside this range are equivalent to angles within it. However, when during the course of a calculation the angle passes out of this range and is then 'reset' (by adding or subtracting 2π), this shows up in the results for Δθ as a discontinuous (and distrcting) jump. Repeat the calculation of Δθ as in Figure 4.19, but do not restrict the value of θ. This should remove the large (Δθ ～ 2π) jumps in Δθ in Figure 4.19, but the smaller and more frequent dips will remain. What is the origin of these dips? Hint: Consider the behavior of a pendulum near one of its turning points.

2. Background

§ 2.1 Three-Body Problem

In physics and classical mechanics, the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with the laws of classical mechanics (Newton's laws of motion and of universal gravitation). The three-body problem is a special case of the n-body problem.

Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth and the Sun. In an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles.

Three body problem

Three Body

Three Body prime

§ 2.2 Ring System

A ring system is a disc or ring orbiting an astronomical object that is composed of solid material such as dust and moonlets, and is a common component of satellite systems around giant planets. A ring system around a planet is also known as a planetary ring system.

The most prominent planetary rings in the Solar System are those around Saturn, but the other three giant planets (Jupiter, Uranus, and Neptune) also have ring systems. Recent evidence suggests that ring systems may be found around other types of astronomical objects, including minor planets, moons, and brown dwarfs.

● Saturn
Saturn's rings are the most extensive ring system of any planet in the solar system, and thus have been known to exist for quite some time. Galileo Galilei first observed them in 1610, but they were not accurately described as a disk around Saturn until Christiaan Huygens did so in 1655. The rings are not a series of tiny ringlets as many think, but are more of a disk with varying density. They consist mostly of water ice and trace amounts of rock, and the particles range in size from micrometers to meters.

Saturn in natural colors (captured by the Hubble Space Telescope)

Saturn's rings dark side mosaic

§ 2.3 Hyperion

Hyperion (/haɪˈpɪəriən/; Greek: Ὑ π ε ρ ί ω ν), also known as Saturn VII (7), is a moon of Saturn discovered by William Cranch Bond, George Phillips Bond and William Lassell in 1848. It is distinguished by its irregular shape, its chaotic rotation, and its unexplained sponge-like appearance. It was the first non-round moon to be discovered.

Hyperion true

3. Main Body

§ 3.1 Claculation

To simulate the motion of Hyperion we will first make few simplifying assumptions.Our goal will not be to perform a relastic simulations.Rather,our objective is simply to show that the motion of such an irregularly shaped moon can be chaotic.With that goal in mind we consider the model with two bodies.We have two particles m1 and m2,connected by a massless rod in orbit around a massive object located at the origin. There are two forces acting on each of the masses,the force of gravity from Saturn and the force from the rod.Since we are interested in the motion about the center of mass,the force from the rod does not contribute.

The coordinateed of the center of mass are (xc , yc), so that (x1 - xc) i + (y1 - yc) j is the vector from the center of mass to m1. The torque on m1 is then:

With a similiar expression for τ2. The total torque on the moon is just τ1 + τ2, and and this is related to the time derivtive of ω by:

where I = m1 r1 ^2 + m2 r2 ^2is the moment of inertia.Putting this all together yields, after some algebra.

where rc is the distance from the center fo mass to Saturn

§ 3.2 Algorithm

Euler_Cromer Method

§ 3.3 Results

For simplicity we took the unit of length to be the radius of Hyperion's orbit (which might be called 1 HU = "Hyperion unit"), and that of time to be the orbital period of Hyperion's around Saturn (1 "Hyperion-year"). Thus, just as in the Earth-Sun case, we have GMSat = 4π^2 in these units. The time step was 0.0001 Hyperion - year.

† 3.3.1 Problem 4.19

‡ Results of the tumbing of Hyperion calculated assuming a particular orbit

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Figure 1: initial speed = 2π, initial θ = 0, so the eccentricity is 1 and the orbit is circular.

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Figure 2: initial speed = 2π, initial ω = 0, so the eccentricity is 1 and the orbit is circular.

The abrupt vertical jumps in θ are simply due to the program "resetting" θ to keep it in the range -π to π (as we did in our pendulum simulations). The behavior in Figure 1 and 2 is seen to be regular and repeatable; this is especially clear from the results for ω. We thus conclude that the motion is not chaotic when the orbit is circular.

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Figure 3: Phase plot for tunbling of Hyperion calculated assuming a circular orbit.

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Figure4: initial speed = 5, initial θ = 0, so the eccentricity > 1 and the orbit is elliptical.

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Figure5: initial speed = 5, initial ω = 0, so the eccentricity > 1 and the orbit is elliptical.

The results obtained for an elliptical orbit, Figure 4 and 5, are very different. The behavior seen in this case is very complicated and erratic, and certainly appears to be chaotic.

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Figure 6: Phase plot for tunbling of Hyperion calculated assuming a eliiptical orbit.

‡ Results of divergence of two nearby trajectories of tumbling motion of Hyperion

I plot the difference between two calculated results for θ(t) with different initial conditions. I used θ(0) = 0 for one trajectory and θ(0) = 0.01 for the other. In all cases the initial ω was zero.

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Figure 7: Calculated for a circular orbit (as considered in Figure 1, 2 and 3).

Figure 8: Scatter diagram of Figure 7.

If I choose an appropriate range for Δθ, we will obtain a better-looking figure, like: set Δθ from 0.0001 to 0.1.

Figure 9: Different Δθ range from Figure 8.

In this circular case we see that while Δθ oscillates some with time, its overall magnitude grows only very slowly. Hence, these two trajectories, θ1(t) and θ2(t), stay near each other, and the motion is not chaotic (as we have already concluded).

Let us take a much closer look at the strange points highly above the other points in Figure 7 that leads to the strange lines in Figure 7. What we need to do is to zoom in an vicinity of one of these strange points in Figure 7.

**From the figure above we can see, In fact, they are not single points, but s series of points and any one is very close to each other. **

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Figure 10: Calculated for a elliptical orbit (the same ellipse as used in Figure 4, 5 and 6).

Figure 11: Scatter diagram of Figure 10.

Figure 12: Different Δθ range from Figure 11.

In contrast, we see that Δθ for elliptical orbit grows rapidly, approximately exponentially, with time until it reaches a value of order π, and it can't get any larger than that. As we saw in Chapter 3, this extrme sensitivity to initial conditions is one of the hallmarks of chaotic behavior.

† 3.3.1 Problem 4.20

If I do not restrict the value of θ. In other words, I remove the following code from my program:

        while self.theta[i + 1] > math.pi:
            self.theta[i + 1] = self.theta[i + 1] - 2 * math.pi
        while self.theta[i + 1] <= -math.pi:
            self.theta[i + 1] = self.theta[i + 1] + 2 * math.pi

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Figure 13: Calculated for a circular orbit wihout restriction of the value of θ.

Figure 14: Calculated for a circular orbit wihout restriction of the value of θ and change time range from [0, 10yr] to [0, 100yr].

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Figure 15: Calculated for a elliptical orbit wihout restriction of the value of θ.

Figure 16: Calculated for a elliptical orbit wihout restriction of the value of θ and change time range from [0, 10yr] to [0, 100yr].

4. Conclusion

† 4.1 Problem 4.19

• The Lyapunov varies as a function of the eccentricity of the orbit, as this following figure shows:
  

The behavior of Δθ versus time (representing the Lyapunov exponent) varies with the change of the eccentricity.

• I can't give the analytical form of function of the exponent with respect to the eccentricity.

‡ 4.2 Problem 4.20

With non-resetting program,the vertical lines in the plot vanish.It is reasonable due to the fact that the 'sudden turning points vanish'.And the θ becomes a continuous vaiable versus time.

5. Acknowledgement

• Prof. Cai
• Wikipedia
• Baidu
• ZZT (Zhang Zitong)